Abstract: We prove that in the fast rotating regime, the three dimensional Gross-Pitaevskii energy describing the state of a Bose Einstein condensate can be reduced to a two dimensional problem and that the vortex lines are almost straight. Additionally, we prove that the minimum of this two dimensional problem can be sought in a reduced space corresponding to the first eigenspace of an elliptic operator. This space is called the Lowest Landau level and is of infinite dimension.